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Problem 1. The following matrix A is diagonalizable. method. −3 −8 A= −4 −8 −4 −10 Problem 2. The following matrix is nilpotent. 0 0 N = 0 0 Find eAt by the diagonalization 10 11 13 Find etN . 1 0 0 0 1 0 . 0 0 1 0 0 0 Problem 3. The following matrix J is in Jordan Canonical Form. Find the matrices S and N so that J = S + N , S is diagonalizable (diagonal, in this case), N is nilpotent, and SN = N S. Compute eJt = eSt eN t by computing etS , etN and multiplying. 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 J = 0 0 0 2 1 0 0 0 0 0 2 0 0 0 0 1 0 0 3 Problem 4. Consider the matrix 568 −347 −31 1257 −1079 −215 791 −485 −61 1767 −1504 −312 −681 419 45 −1517 1299 262 A= . −620 393 37 −1393 1209 229 −686 432 46 −1542 1331 259 129 −60 −6 261 −204 −59 A. Find a basis of each of the generalized eigenspaces of A. B. Find the matrices S and N so that A = S + N , S is diagonalizable, N is nilpotent, and SN = N S. Verify that your answers for S and N really satisfy these conditions. (It is not necessary to go all the way to the Jordan Canonical Form of A.) C. Use S and N to compute eAt = etS etN . 2 Problem 5. In each part, use Leonard’s Algorithm to find eAt . A. 5 −1 A= 12 −2 3 −1 B. 0 0 2 1 1 −2 A= −3 5 −6 0 1 −1 C. 1 1 −2 −3 5 −6 A= 0 1 −1 D. −79 −4 −36 614 7 A= 262 4 71 −2 76 −558 137 −240 43 −61 325 Problem 6. Recall that we have shown et(A+B) = etA etB provided that A and B commute. To see what happens if A and B do not commute, consider the function Ψ(t) = et(A+B) e−tB e−tA , so if A and B commute, Ψ(t) = I for all t. To see how much Ψ differs from the identity in the general case, find the first two nonzero terms in the Taylor expansion of Ψ(t), centered at t = 0. Problem 7. Consider the following mechanical system. 3 We have two weights of masses m1 and m2 connected by three springs to a stationary frame. The spring constants of the three springs are k1 , k2 and k3 , as indicated in the diagram. The weights are constrained to move in a horizontal line. The deviation of the two weights from their equilibrium positions are y1 and y2 respectively, measured positive to the right. A. Assuming there is no friction, write down the system of linear equations that governs the motion of the weights, using Hooke’s law for the force produced by the springs. By very careful with your signs! Make this system into a first order system by introducing new dependent variables z1 and z2 and adding the equations y10 = z1 and y20 = z2 (see Section 3.1 in the book). Write the resulting system in matrix form dy/dt = Ay. B. Assume the values m1 = 1, m2 = 2, k1 = 1, k2 = 2, k3 = 2. Find eAt , where A is the coefficient matrix of the system, using one of the methods developed in class. Find the solution for the initial conditions y1 (0) = 1, y2 (0) = −1, y10 (0) = 0, y20 (0) = 0. Provide a graph of y1 and y2 as functions of time. C. Assume now that the weights are subject to frictional forces proportional to their velocities, i.e., the force on the first weight due to friction is −b1 y10 and the force on the second weight due to friction is −b2 y20 , where b1 and b2 are positive constants. Find the new system of equations dy/dt = By. D. Assume the values m1 = 1, m2 = 1 k1 = 1, k2 = 1, k3 = 1 b1 = 1/2, b2 = 1/2. Find eBt , using one of the methods developed in class. Solve the system subject to the initial conditions y1 (0) = 1, y2 (0) = 0, y10 (0) = 0 y20 (0) = 0. Provide a graph showing y1 and y2 as functions of time. 4 Problem 8. Do Problem 20 on page 190 of the book. See Section 1.7 in the book for how to model electric circuits. 5 EXAM Practice Exam # 2 Math 3351, Spring 2003 March 30, 2003 • Write all of your answers on separate sheets of paper. You can keep the exam questions when you leave. You may leave when finished. • You must show enough work to justify your answers. Unless otherwise instructed, give exact answers, not √ approximations (e.g., 2, not 1.414). • This exam has 8 problems. There are 0 points total. Good luck!